This page presents the general trajectories of photons around a Kerr black hole, based on the analysis of the roots of the radial effective potential \(V_r\) — a 4th degree polynomial in \(r\), the photon radial coordinate. Depending on the values of the constants of motion, this potential can have up to four positive or zero real roots, defining different types of photon trajectories: deflection, constant radius orbit, or absorption by the black hole, with the possibility of transition to another region of spacetime. Different graphs of \(V_r\) are analyzed, and a descriptive table summarizes all possible patterns. The case of the over-extreme Kerr object is also discussed.
Contents
INTRODUCTION
The behavior of the radial coordinate \(r\) of the photon is described by the effective radial potential, which is written as: \(\frac{c^2}{\varepsilon^2}V_r=r^4+\left(a^2-c^2\frac{l_z^2}{\varepsilon^2} -c^2\frac{Q}{\varepsilon^2}\right)r^2+2m\left(\left(a-c\frac{l_z}{\varepsilon}\right)^2+c^2\frac{Q}{\varepsilon^2}\right)r-a^2c^2\frac{Q}{\varepsilon^2}\)
and which must be positive or zero throughout the photon trajectory (refer to the axial symmetry study for definitions of terms).
Depending on the values of \(a\), \(l_z\), and \(Q\), this 4th-degree polynomial has 0, 1, or several positive or zero real roots, which can be simple, double, triple or quadruple.
RADIAL EFFECTIVE POTENTIAL – Kerr black hole
The most common case for the radial effective potential \(V_r\) is a single positive root greater than \(r_h\) (radial coordinate of the event horizon), which causes a simple deflection of the photon due to repulsivity: when the radial coordinate of a photon entering from \(+\infty\) reaches this value, the photon “rebounds” (change in the sign of variation of \(r\)) and exits towards \(+\infty\).
The other cases are related to the roots multiplicity:
– a non-zero double positive root corresponds to the cancellation of \(\frac{dVr}{dr}\): when the radial coordinate of a photon reaches this value, the photon joins an orbit with coordinate \(r_c\) equal to the value of the double root,
– a non-zero positive triple root corresponds to the cancellation of \(\frac{dVr}{dr}\) and \(\frac{d^2Vr}{dr^2}\): when the radial coordinate of a photon reaches this value, the photon joins the stability limit orbit with a coordinate \(r_c\) equal to \(r_{c_{stab}}\), the value of the triple root,
– the zero quadruple root means that the photon reaches the singularity of the black hole: the central point \(r=0\) for a Schwarzschild black hole or the annular orbit \(r_c=0\) for a black hole with a non-zero Kerr parameter \(a\).
The figures below highlight different possible cases for a photon coming from the asymptotic region (entering from \(+\infty\)), using, for example, \(\bar{a}=0.95\) and arbitrary values of \(l_z\) and then \(Q\), except for the triple root and quadruple root, which force these values.
Examples of actual trajectory and orbit plots are provided in the axial symmetry study.
Refer also to the description table – Kerr black hole below for all possible patterns for an incoming or outgoing photon, whether from \(+\infty\) or not.
Note 1: there is no root of the potential \(V_r\) between the event horizon and Cauchy horizon, as this would imply a change in the direction of variation of \(r\), contradicting the temporal nature of the radial coordinate \(r\) between the two horizons.
Note 2: the photon motion is only defined for positive or zero radial potential \(V_r\) and colatitude potential \(V_{\theta}\). There are therefore “non-admissible” domains \((r, \theta)\) for given values of \(a\), \(l_z\), and \(Q\): the photon cannot have a radial coordinate \(r\) such that \(V_r<0\) or a colatitude \(\theta\) such that \(V_\theta=Q+\cos^2\theta\left(a^2\frac{\varepsilon^2}{c^2}-\frac{l_z^2}{\sin^2\theta}\right)<0\), it being understood that \(\theta=90^\circ\) is always an admissible value for positive or zero values of Carter constant \(Q\).
no root > 0: the photon reaches the central disk (\(r=0\) and \(\theta\ne\frac{\pi}{2})\) and enters negative space \(r<0\).
1 simple root < \(\bar{r}_{Cauchy}\) and 1 double root > \(\bar{r}_h\) -> the photon reaches the unstable orbit \(\bar{r}_c\simeq 1.392\):
– an external instability sends it back to \(+\infty\),
– an internal instability causes it to bounce off the single root \(\bar{r}_{lim}\simeq 0.034\) and sends it back to the event horizon, exiting the initial universe.
1 simple root and 1 double root < \(\bar{r}_{Cauchy}\) -> the photon reaches the unstable orbit \(\bar{r}_c\simeq 0.647\):
– an external instability sends it back to the Cauchy horizon and then to the event horizon, exiting the initial universe,
– an internal instability causes it to bounce off the single root \(\bar{r}_{lim}\simeq 0.107\) and sends it back to the unstable orbit (the photon then remains trapped between the two roots).
1 double root and 1 simple root < \(\bar{r}_{Cauchy}\) -> the photon bounces off the simple root \(\bar{r}_{lim}\simeq 0.663\), which sends it back toward the Cauchy horizon and then the event horizon, exiting the initial universe.
If a photon is emitted very precisely at the value of the double root \(\bar{r}=\bar{r}_c\simeq 0.256\) with a value \(\theta\) such that \(V\theta\ge 0\), it remains indefinitely on the stable orbit \(\bar{r}_c\).
1 triple root = \(\bar{r}_{c_{stab}}\) less than \(\bar{r}_{Cauchy}\) -> the photon reaches the stability limit orbit \(\bar{r}_{c_{stab}}\simeq 0.540\):
– an external instability sends it back to the Cauchy horizon and then to the event horizon, exiting the initial universe,
– otherwise it remains in the orbit \(\bar{r}_{c_{stab}}\).
3 simple roots > 0 with 1 less than \(\bar{r}_{Cauchy}\) and 2 greater than \(\bar{r}_h\) – >the photon bounces off the greatest root \(\bar{r}_{lim3}\simeq 1.673\), which causes it to exit towards \(+\infty\).
If an incoming photon is emitted with \(r\) between the root \(\bar{r}_{lim1}\simeq 0.248\) and the root \(\bar{r}_{lim2}\simeq 1.335\) and a value \(\theta\) such that \(V_\theta\ge 0\), it bounces off \(\bar{r}_{lim1}\), which sends it back towards the event horizon exiting the initial universe.
3 simple roots > 0 and less than \(\bar{r}_{Cauchy}\): the photon bounces off the largest root \(\bar{r}_{lim3}\simeq 0.685\), which sends it back toward the Cauchy horizon and then the event horizon, exiting the initial universe.
If a photon is emitted with \(r\) between the root \(\bar{r}_{lim1}\simeq 0.225\) and the root \(\bar{r}_{lim2}\simeq 0.499\) and a value \(\theta\) such that \(V_\theta\ge 0\), it remains trapped between these two roots.
1 quadruple zero root -> the photon reaches the singularity of the black hole:
– the central point \(r=0\) following a radial trajectory for a Schwarzschild black hole, or
– the unstable annular orbit \(r_c=0\) following an equatorial trajectory for a black hole with a non-zero Kerr parameter \(a\).
In the latter case, an external instability sends it back to the Cauchy horizon and then to the event horizon exiting the initial universe, and an internal instability causes it to enter negative space \(r<0\).
DESCRIPTION TABLE – Kerr black hole
The table below lists the different possible cases (\(V_r\ge 0)\) for an incoming or outgoing photon, initially located in an admissible domain.
The cases \(\bar{a}\in[-1, 0]\) are obtained by replacing \(\bar{a}\) and \(l_z\) with \(-\bar{a}\) and \(-l_z\), respectively.
The meaning of the abbreviations is given below the table.
Abbreviations used:
ā: Kerr parameter
domain r0: admissible domain for initial radial coordinate (radial potential Vr >= 0)
I: incoming photon (decreasing r)
lz: component of the photon angular momentum on the spin axis of the Kerr black hole (0 < ā <=1)
lzstab: value of lz for the stability limit orbit rc = rcstab (ā ≠ 0)
lzx-: value of lz for the unstable prograde equatorial orbit rc > rh (ā ≠ 0)
lzx+: value of lz for the unstable retrograde equatorial orbit rc > rh (ā ≠ 0)
lzxin: value of lz for the stable prograde equatorial orbit rc < rCauchy (ā ≠ 0)
O: outgoing photon (increasing r)
Q: Carter constant
Qcrit-: value of Q for the unstable orbit rc < rCauchy (ā ≠ 0) corresponding to the value lz
Qcrit+: value of Q for the unstable orbit rc > rh (ā ≠ 0) corresponding to the value lz
Qcritstab: value of Q for the stable orbit rc < rcstab < rCauchy (ā ≠ 0) corresponding to the value lz
Qstab: value of Q for the stability limit orbit rc = rstab (ā ≠ 0) corresponding to the value lzstab
r0: initial radial coordinate of the photon
rc: constant orbital radial coordinate
rCauchy: radial coordinate of the Cauchy horizon (ā ≠ 0)
rcstab: constant radial coordinate of the stability limit orbit (ā ≠ 0)
rh: radial coordinate of the event horizon
rlim: positive or zero root of the radial potential
RADIAL EFFECTIVE POTENTIAL – Over-extreme Kerr object
A Kerr parameter such as \(|\bar{a}|>1\) leads to an over-extreme Kerr object with no horizons. The existence of such an astrophysical object is unlikely, so the following paragraphs are mathematical descriptions.
The radial effective potential \(V_r\) of an over-extreme Kerr object object has the same type of roots as those described for the Kerr black hole (see above).
The only difference is the absence of a Cauchy horizon and an event horizon, which in particular prevents photons from leaving the initial universe.
The figures below are equivalent to those presented above for different possible cases of a photon coming from the asymptotic region (entering from \(+\infty\)), using, for example, (\bar{a}=1.5) and arbitrary values of \(l_z\) and then \(Q\), except for the triple root and quadruple root, which force these values.
Refer also to the description table – over-extreme Kerr object below for all possible behaviors for an incoming or outgoing photon, from \(+\infty\) or not.
The admissible domain \((r,\theta)\) for given values \(a\), \(l_z\) and \(Q\) is defined as for the Kerr black hole, namely by \(V_r\ge 0\) and \(V_\theta\ge 0\).
no root > 0: the photon reaches the central disk (\(r=0\) and \(\theta\ne\frac{\pi}{2})\) and enters negative space \(r<0\).
1 simple root and 1 double root -> the photon reaches the unstable orbit \(\bar{r}_c=3\):
– an external instability sends it back to \(+\infty\),
– an internal instability causes it to bounce off the single root \(\bar{r}_{lim}\simeq 0.967\) and sends it back to the unstable orbit (the photon then remains trapped between the two roots).
1 double root and 1 simple root -> the photon bounces off the simple root \(\bar{r}_{lim}\simeq 5.454\), which sends it back to \(+\infty\).
If a photon is emitted very precisely at the value of the double root \(\bar{r}=\bar{r}_c\simeq 1.413\) with a value \(\theta\) such that \(V\theta\ge 0\), it remains indefinitely on the stable orbit \(\bar{r}_c\).
1 triple root = \(\bar{r}_{c_{stab}}\) -> the photon reaches the stability limit orbit \(\bar{r}_{c_{stab}}\simeq 2.077\):
– an external instability sends it back to \(+\infty\),
– otherwise it remains in the orbit \(\bar{r}_{c_{stab}}\).
3 simple roots > 0: the photon bounces off the greatest root \(\bar{r}_{lim3}\simeq 4.940\), which causes it to exit towards \(+\infty\).
If an incoming photon is emitted with \(r\) between the root \(\bar{r}_{lim1}\simeq 1.022\) and the root \(\bar{r}_{lim2}\simeq 1.968\) and a value \(\theta\) such that \(V_\theta\ge 0\), it bounces off \(\bar{r}_{lim1}\), and the photon then remains trapped between the two roots.
1 quadruple zero root -> the photon reaches the singularity of the black hole:
– the central point \(r=0\) following a radial trajectory for a Schwarzschild black hole, or
– the unstable annular orbit \(r_c=0\) following an equatorial trajectory for a black hole with a non-zero Kerr parameter \(a\).
In the latter case, an external instability sends it back to \(+\infty\), and an internal instability causes it to enter negative space \(r<0\).
DESCRIPTION TABLE – Over-extreme Kerr object
The table below lists the different possible cases (\(V_r\ge 0\)) for an incoming or outgoing photon, initially located in an admissible domain.
The cases \(\bar{a}\in]-\infty, -1[\) are obtained by replacing \(\bar{a}\) and \(l_z\) with \(-\bar{a}\) and \(-l_z\), respectively.
The meanings of the abbreviations are given below the table.
Abbreviations used:
ā: Kerr parameter
domain r0: admissible domain for initial radial coordinate (radial potential Vr >= 0)
I: incoming photon (decreasing r)
lz: component of the photon angular momentum on the spin axis of the over-extreme Kerr object
lzstab: value of lz for the stability limit orbit rc = rcstab
lzx+: value of lz for the unstable retrograde equatorial orbit
O: outgoing photon (increasing r)
Q: Carter constant
Qcrit: value of Q for the unstable orbit rc > rcstab corresponding to the value lz
Qcritstab-: value of Q for the stable orbit rc < 1 corresponding to the value lz
Qcritstab+: value of Q for the stable orbit 1 < rc < rcstab corresponding to the value lz
Qstab: value of Q for the stability limit orbit rc = rstab corresponding to the value lzstab
r0: initial radial coordinate of the photon
rc: constant orbital radial coordinate
rcstab: constant radial coordinate of the stability limit orbit
rlim: positive or zero root of the radial potential